Solve for x in the equation involving square roots

Published on April 11, 2025

Algebra

The solutions to the equation \sqrt{5x^2-6x+8} - \sqrt{5x^2-6x-7} = 1 are x = 4 and x = -14/5.

Question

Answer

We are asked to solve for

in the equation:

Let

. The equation becomes:

To solve this equation, we isolate one of the square roots and square both sides.

Squaring both sides, we get:

Subtract

from both sides:

Add 6 to both sides:

Divide by 2:

Square both sides:

Add 7 to both sides:

Now substitute back

We use the quadratic formula to solve for

Here,

We find that

We have two solutions:

We check both solutions in the original equation.
For

For

Both solutions are valid.

Final Answer: The final answer is

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